Step 1 Trim the wings and backbone off the chicken along with any excess fat. Season the pieces well on both sides with salt and pepper. Heat the oil and butter in a Dutch oven or deep heavy skillet over medium-high heat. Dredge the chicken pieces in flour to coat both sides, then shake off the excess. Working in batches if necessary, add the chicken parts to the pan, skin side down, and cook until golden brown, about 5 minutes on a side. Remove to a platter.
Step 2 Pour off all but 1 tablespoon of the fat from the pan. Lower the heat to medium and add the Calvados, stirring to scrape up any crusty bits off the bottom (it may flame briefly). Add the shallots and thyme and cook until the liquid is reduced to a glaze, about 1 minute. Return the chicken to the pan, skin side up, and pour the cider around the pieces. Bring to a simmer, cover and cook 20 minutes, or until the juices run clear when the pieces are pricked. (The legs may take a few minutes longer.)
Step 3Remove the chicken pieces to a platter and keep warm. Raise the heat under the pan and add the apple pieces. Cook, stirring, until the apples are soft but not mushy and the sauce is thickened, about 3 to 4 minutes. Taste and adjust the seasoning.
Step 4Divide the chicken pieces among serving plates and spoon the sauce and apples alongside.